A shop sells 200 mugs a week at $10 each. For every $1 it adds to… Step by Step

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The question

Question: A shop sells 200 mugs a week at $10 each.

For every $1 it adds to the price, it sells 10 fewer mugs.

What price gives the most money?

1 Draw and name

Call the number of one-dollar rises x.

x = the number of $1 rises

Then the price is $10 + x.

price = 10 + x

And each rise loses 10 mugs: they sell 200 − 10x.

mugs = 200 − 10x

2 Write the money

The money is the price times the number sold.

R = price × number sold

R = (10 + x)(200 − 10x)

Multiply out the brackets.

Now we have R = 2000 + 100x − 10x².

= 2000 + 100x − 10x²

3 Which x make sense

x is 0 or more, and they must still sell some mugs: x is less than 20.

0 ≤ x < 20 =

4 Slope = 0

The biggest value is where the slope is 0.

Use the power rule.

R′(x) = 0 + 100 × 1 − 10 × 2x^(2 − 1)

Now we have R′(x) = 100 − 20x.

= 100 − 20x

At a turning point the graph is flat for a moment: the slope is 0.

So we solve R′(x) = 0.

R′(x) = 0

100 − 20x = 0

Take out the −20.

−20(−5 + x) = 0

So x = 5.

x = 5

5 Make sure it is the biggest

Before 5, the slope is positive: going up.

After it, negative: going down.

Up, then down: the top.

A maximum.

R′(4) = 20 > 0 ↗, R′(6) = −20 < 0 ↘

6 Answer the question

The best price is $10 + 5: $15.

price = 10 + 5 = $15

Careful!

x is the number of one-dollar rises, not the price.

The price is 10 + 5.

Careful: not price = $5 ✗

Then they sell 150 mugs, and take in $2250.

R(5) = 15 × 150 = $2250

7 Check

Let's check with x a bit smaller: 4 gives 2240.

x = 4: R = 2240

And x = 6 gives 2240.

Both are smaller than 2250.

Right!

x = 6: R = 2240

Let's look back.

First: draw and name.

Then: write the money.

Then: which x make sense.

Then: slope = 0.

Then: make sure it is the biggest.

Then: answer the question.

Last: check.

Answer